Regularity and permutability via transferability of tolerances. (English) Zbl 0957.08001

Summary: An algebra \(A\) has \(n\)-transferable tolerances if for any \(a,b,c\in A\) there exist \(d_1,..., d_n \in A\) such that \(T(a,b)= T(c,d_1)\vee \dots\vee T(c,d_n)\) in the tolerance lattice \(\operatorname {Tol}A\). We prove that a variety \({\mathcal V}\) is regular and permutable if and only if each \(A\in{\mathcal V}\) has \(n\)-transferable tolerances. Analogously we characterize varieties of 0-regular and permutable congruences.


08A30 Subalgebras, congruence relations
08B05 Equational logic, Mal’tsev conditions
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